"p(x)=(x^3-1,2x^4+2x^3+7x^2+5x+5)" is the greatest common divisor of
"x^3-1" and "2x^4+2x^3+7x^2+5x+5" .
Find "p(x)" according to Euclid's algorithm
"2x^4+2x^3+7x^2+5x+5:x^3-1"
"\\begin{matrix} 2x^4+2x^3+7x^2+5x+5 &&&& |x^3-1 \\\\ \n2x^4-2x &&&& 2x+2\\\\ \n&&2x^3+7x^2+5x+5\\\\\n &&2x^3-2&\\\\\n &&&7x^2+7x+7\\\\ \n&&&x^2+x+1 \\end{matrix}"
"x^3-1:x^2+x+1"
"\\begin{matrix} x^3-1 &&& |x^2+x+1 \\\\ x^3+x^2+x &&&x-1 \\\\ & -x^2-x-1\\\\ &-x^2-x-1\\\\ &&0 \\end{matrix}"
Then "p(x)=x^2+x+1" .
"Q[x]\/p(x)=Q[x] \/<x^2+x+1>" .
The set "Q[x]\/p(x)" the product of polynomials from "Q[x]" by "p(x)=x^2+x+1" .
We are obsessed with polynomials "Q[x]" .
Then "Q[x]\/ p(x)=Q[x]\/Q[x]=\\{0\\}"
It is not a field, because there "x^2+x+1=0"
Answer: "p(x)=x^2+x+1" , "Q[x]\/I" is not a field
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Dear Ram, please use the panel for submitting new questions.
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Dear Ram, please use the panel for submitting new questions.
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